4-4 Rational Numbers

Property

A rational number is a number that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. All fractions, both positive and negative, are rational numbers.
Since any integer, terminating decimal, or repeating decimal can be written as a ratio of two integers, they are all rational numbers.

Examples

• To write the integer $-25$ as a ratio of two integers, express it as a fraction with a denominator of 1: $\frac{-25}{1}$.
• The decimal $9.37$ can be written as a mixed number $9\frac{37}{100}$, which converts to the improper fraction $\frac{937}{100}$.
• The mixed number $-4\frac{2}{3}$ is equivalent to the improper fraction $-\frac{14}{3}$.

Explanation

Think of 'rational' as 'ratio-nal.' Any number that can be expressed as a simple fraction or ratio between two integers is a rational number. This includes whole numbers, integers, and decimals that either end or repeat predictably.

Property

To represent the rational number system by points on a line, first draw a horizontal line and mark a point as the origin, denoted by $0$. Mark off a succession of equally spaced points to the right of $0$ as $1, 2, 3, \ldots$ and to the left as $-1, -2, -3, \ldots$.
To place a fraction like $p/q$, divide the unit interval into $q$ equal parts.
The point representing $p/q$ is found by taking $p$ of these parts, to the right of the origin if $p/q$ is positive, and to the left if negative.

Examples

• The fraction $-\frac{3}{4}$ is located by dividing the segment from $0$ to $-1$ into four equal parts and marking the point three parts to the left of the origin.
• The fraction $\frac{2}{5}$ is located by dividing the segment from $0$ to $1$ into five equal parts and marking the end of the second part.
• The fraction $-\frac{1}{2}$ is located at the midpoint of the interval between $0$ and $-1$.

Explanation

A number line gives every rational number a unique address. Integers are evenly spaced markers, and fractions are the points in between, located by dividing the spaces into equal parts. This helps us visualize the value and order of numbers.

Property

The absolute value $|a|$ of a rational number $a$ is the distance from the point on the line to $0$.
A number and its opposite have the same absolute value.
Every rational number has an opposite, or additive inverse. $0$ is its own opposite.

Examples

• The absolute value of $-15$ is its distance from $0$, so $|-15| = 15$.

• The opposite of $9.5$ is $-9.5$. Both numbers are $9.5$ units from $0$, so $|9.5| = |-9.5| = 9.5$.